Fading is one of the major problems in communication systems. It represents random fluctuations in the amplitude of the received signal due to multi-path propagation. If the delay spread of the channel is larger than the symbol period of the signal, the fading is also frequency selective. The amplitude of fading is usually approximated by a Rayleigh distribution. Such fading is referred to as Rayleigh fading.
In digital communication systems, information is encoded as a sequence of symbols belonging to a discrete alphabet, referred to as a constellation. Such a constellation has N dimensions and encodes B information bits per dimension. The number of possible values, also referred to as constellation points, is therefore 2N*B. The number of bits per dimension B directly determines the spectral efficiency of the transmission, given in bits/Hz. The number of dimensions N has no effect on the spectral efficiency. An example constellation with N=2 and B=1 is illustrated in FIG. 1A.
Traditionally, for example in a quadrature amplitude modulation (QAM) constellation shown in FIG. 1A, each transmitted bit affects only one dimension. Referring to FIG. 1A, “b1” of each constellation point “b1b2” (=“00”, “01”, “10” and “11”) affects only the dimension represented by the horizontal axis, whereas “b2” of each constellation point “b1b2” affects only the dimension represented by the vertical axis. If the dimension affected by the transmitted bits undergoes a deep fading, all bits that modulate this dimension will be extremely unreliable, which increases the error probability. This effect is illustrated by the errors in FIG. 1A. For example, if the channel represented by the vertical axis fades away, the constellation points “00”, “01”, “10” and “11” will approach the horizontal axis (along the solid arrows of FIG. 1A). As a result, the constellation points “00” and “01”, as well as the constellation points “10” and “11”, will be indiscernible.
If the constellation is modified such that each bit affects all dimensions, the resilience to fading is increased. A deep fading on one of the dimensions will affect all the bits of the constellation; however, this effect would not be as detrimental as in the conventional case, so that on average, the error probability decreases. This is referred to in the literature as modulation diversity.
(Rotated Constellations)
One way to achieve modulation diversity is to rotate a (hyper-cubic) constellation to spread the effect of a channel fading over all its dimensions. This is illustrated in FIG. 1B for the case where N=2 and B=1. For example, as shown in FIG. 1B, if the channel represented by the vertical axis fades away, the constellation points “00”, “01”, “10” and “11” will approach the horizontal axis (along the solid arrows of FIG. 1B). However, these constellation points will still be discernible in the dimension represented by the horizontal axis. As such, the constellation points “00”, “01”, “10” and “11” remain discernible even after a deep fading of the channel represented by the vertical axis.
A multi-dimensional rotation can be achieved by multiplying the N-element signal vector by an N*N square matrix. The necessary and sufficient condition for a square matrix to be a rotation matrix (or a reflection matrix) is for it to be orthogonal, i.e., to satisfy the equation of the following Math. 1.T=I  [Math. 1]
Note that in the above Math. 1, the matrix    is a square matrix, the matrix    T is a transpose matrix of the matrix    ,and the matrix    is a unit matrix.
This means that with regard to the above Math. 1, the row/column vectors must be orthogonal unit vectors, i.e., satisfy the equation of the following Math. 2.
                                          ∑                          i              =              1                        n                    ⁢                                          ⁢                                    r                              i                ,                j                                      ⁢                          r                              i                ,                k                                                    =                  δ                      j            ,            k                                              [                  Math          .                                          ⁢          2                ]            
Note that in Math. 2,                δj,k=1if        j=k ,and        δj,k=0if        j≠k.        
This preserves the Euclidean distance between any two points of the constellation, and ensures that the performance in channels with additive white Gaussian noise (AWGN channels) is not affected.
Obviously, not all rotations yield the effect of improved modulation diversity. From NPL 1, it is known that the optimum rotation angle    θfor 16-QAM satisfies the equation shown in the following Math. 3. The corresponding 2-D (two-dimensional) rotation matrix    satisfies the equation shown in the following Math. 4.
                    θ        =                  π          /          8                                    [                  Math          .                                          ⁢          3                ]                                R        =                  (                                                                      cos                  ⁢                                                                          ⁢                  θ                                                                                                  -                    sin                                    ⁢                                                                          ⁢                  θ                                                                                                      sin                  ⁢                                                                          ⁢                  θ                                                                              cos                  ⁢                                                                          ⁢                  θ                                                              )                                    [                  Math          .                                          ⁢          4                ]            
Finding the optimum rotation for constellations of more than two dimensions is more complicated, because there is no single optimization parameter such as the one pertaining to the rotation angle in a 2-D constellation. In the case of a 4-D (four-dimensional) constellation, for example, there are six independent rotation angles, each with its own partial rotation matrix. The partial rotation angles are also called Givens angles in NPL 2. The final 4-D rotation matrix is obtained by multiplying the six Givens rotation matrices, namely the six matrices shown in the following Math. 5.
                                          R            4                          1              ,              2                                =                      (                                                                                                      +                      cos                                        ⁢                                                                                  ⁢                                          θ                                              1                        ,                        2                                                                                                                                                        -                      sin                                        ⁢                                                                                  ⁢                                          θ                                              1                        ,                        2                                                                                                              0                                                  0                                                                                                                        +                      sin                                        ⁢                                                                                  ⁢                                          θ                                              1                        ,                        2                                                                                                                                                        +                      cos                                        ⁢                                                                                  ⁢                                          θ                                              1                        ,                        2                                                                                                              0                                                  0                                                                              0                                                  0                                                  1                                                  0                                                                              0                                                  0                                                  0                                                  1                                                      )                          ,                                  ⁢                              R            4                          1              ,              3                                =                      (                                                                                                      +                      cos                                        ⁢                                                                                  ⁢                                          θ                                              1                        ,                        3                                                                                                              0                                                                                            -                      sin                                        ⁢                                                                                  ⁢                                          θ                                              1                        ,                        3                                                                                                              0                                                                              0                                                  1                                                  0                                                  0                                                                                                                        +                      sin                                        ⁢                                                                                  ⁢                                          θ                                              1                        ,                        3                                                                                                              0                                                                                            +                      cos                                        ⁢                                                                                  ⁢                                          θ                                              1                        ,                        3                                                                                                              0                                                                              0                                                  0                                                  0                                                  1                                                      )                          ,                                  ⁢                              R            4                          1              ,              4                                =                      (                                                                                                      +                      cos                                        ⁢                                                                                  ⁢                                          θ                                              1                        ,                        4                                                                                                              0                                                  0                                                                                            -                      sin                                        ⁢                                                                                  ⁢                                          θ                                              1                        ,                        4                                                                                                                                          0                                                  1                                                  0                                                  0                                                                              0                                                  0                                                  1                                                  0                                                                                                                        +                      sin                                        ⁢                                                                                  ⁢                                          θ                                              1                        ,                        4                                                                                                              0                                                  0                                                                                            +                      cos                                        ⁢                                                                                  ⁢                                          θ                                              1                        ,                        4                                                                                                                  )                          ,                                  ⁢                              R            4                          2              ,              3                                =                      (                                                            1                                                  0                                                  0                                                  0                                                                              0                                                                                            +                      cos                                        ⁢                                                                                  ⁢                                          θ                                              2                        ,                        3                                                                                                                                                        -                      sin                                        ⁢                                                                                  ⁢                                          θ                                              2                        ,                        3                                                                                                              0                                                                              0                                                                                            +                      sin                                        ⁢                                                                                  ⁢                                          θ                                              2                        ,                        3                                                                                                                                                        +                      cos                                        ⁢                                                                                  ⁢                                          θ                                              2                        ,                        3                                                                                                              0                                                                              0                                                  0                                                  0                                                  1                                                      )                          ,                                  ⁢                              R            4                          2              ,              4                                =                      (                                                            1                                                  0                                                  0                                                  0                                                                              0                                                                                            +                      cos                                        ⁢                                                                                  ⁢                                          θ                                              2                        ,                        4                                                                                                              0                                                                                            -                      sin                                        ⁢                                                                                  ⁢                                          θ                                              2                        ,                        4                                                                                                                                          0                                                  0                                                  1                                                  0                                                                              0                                                                                            +                      sin                                        ⁢                                                                                  ⁢                                          θ                                              2                        ,                        4                                                                                                              0                                                                                            +                      cos                                        ⁢                                                                                  ⁢                                          θ                                              2                        ,                        4                                                                                                                  )                          ,                                  ⁢                              R            4                          3              ,              4                                =                      (                                                            1                                                  0                                                  0                                                  0                                                                              0                                                  1                                                  0                                                  0                                                                              0                                                  0                                                                                            +                      cos                                        ⁢                                                                                  ⁢                                          θ                                              3                        ,                        4                                                                                                                                                        -                      sin                                        ⁢                                                                                  ⁢                                          θ                                              3                        ,                        4                                                                                                                                          0                                                  0                                                                                            +                      sin                                        ⁢                                                                                  ⁢                                          θ                                              3                        ,                        4                                                                                                                                                        +                      cos                                        ⁢                                                                                  ⁢                                          θ                                              3                        ,                        4                                                                                                                  )                                              [                  Math          .                                          ⁢          5                ]            
From NPL 2, it is known that the optimization may be carried out over the vector having the six elements shown in the following Math. 6.θ=(θ1,2,θ1,3,θ1,4,θ2,3,θ2,4,θ3,4)  [Math. 6]
According to NPL 2, the resulting optimum rotation angles for a 4-D constellation with two bits per dimension have the values shown in the following Math. 7.
                                 {                                                                                          θ                                          1                      ,                      2                                                        =                                      39                    ⁢                    °                                                                                                                                            θ                                          1                      ,                      3                                                        =                                      25                    ⁢                    °                                                                                                                                            θ                                          1                      ,                      4                                                        =                                      43                    ⁢                    °                                                                                                                                            θ                                          2                      ,                      3                                                        =                                      53                    ⁢                    °                                                                                                                                            θ                                          2                      ,                      4                                                        =                                      41                    ⁢                    °                                                                                                                                            θ                                          3                      ,                      4                                                        =                                      23                    ⁢                    °                                                                                                            [                      Math            .                                                  ⁢            7                    ]                    
The disadvantage of this method is the number of parameters, specifically for a large number of dimensions. For N dimensions, the number of partial rotation angles is equal to the number of possible combinations of two from a set of N, i.e., the value given by the following Math. 8.
                              (                                                    N                                                                    2                                              )                =                                            N              !                                                      2                !                            ⁢                                                (                                      N                    -                    2                                    )                                !                                              =                                    N              ⁡                              (                                  N                  -                  1                                )                                      2                                              [                  Math          .                                          ⁢          8                ]            
Hence, the number of rotation angles increases with the square of the number of dimensions, so the optimization problem becomes very difficult when the number of dimensions is large.
NPL 3 discloses two different approaches, relying on the use of the algebraic number theory, which have the advantage of a reduced number of parameters.
The first approach allows the construction of rotation matrices by applying the “canonical embedding” to an algebraic number field. Two methods are proposed. The first method produces lattices with diversity L=N/2 for the number of dimensions N=2e23e3, with e2, e3=0, 1, 2, . . . . Diversity means the minimum number of different values in the components of any two distinct points of the constellation. The second method produces lattices with diversity L=N. The possible values of N are very limited, such as 3, 5, 9, 11, and 15.
A variant of this method for generating N-dimensional rotated constellations is also known from NPL 3. The rotation matrix    is expressed by the following Math. 9.
                    R        =                                            2              N                                ⁢                      cos            ⁡                          (                                                                                                                  2                        ⁢                                                                                                  ⁢                        π                                                                    8                        ⁢                                                                                                  ⁢                        n                                                              ⁡                                          [                                                                        4                          ×                                                      [                                                          1                              ,                              2                              ,                              …                              ⁢                                                                                                                          ,                              N                                                        ]                                                                          -                        1                                            ]                                                        T                                ⁡                                  [                                                            2                      ×                                              [                                                  1                          ,                          2                          ,                          …                          ⁢                                                                                                          ,                          N                                                ]                                                              -                    1                                    ]                                            )                                                          [                  Math          .                                          ⁢          9                ]            
Note that the superscripted letter “T” denotes the transpose of a matrix.
For N=4, the value of the rotation matrix    is given by the following Math. 10.
                    R        =                  (                                                                      +                  0.5879                                                                              -                  0.1379                                                                              -                  0.6935                                                                              -                  0.3928                                                                                                      +                  0.1379                                                                              -                  0.3928                                                                              +                  0.5879                                                                              -                  0.6935                                                                                                      -                  0.3928                                                                              +                  0.6935                                                                              -                  0.1379                                                                              -                  0.5879                                                                                                      -                  0.6935                                                                              -                  0.5879                                                                              -                  0.3928                                                                              -                  0.1379                                                              )                                    [                  Math          .                                          ⁢          10                ]            
Although the resulting rotation matrix is a rotation matrix that is orthogonal for any N, the full modulation diversity is only achieved when N is a power of two.
Each of these methods can guarantee a certain degree of diversity. However, the resulting rotation matrix is fixed, having no parameter that allows the optimization for different constellation sizes. Therefore, a severe disadvantage of these methods is that the effect of modulation diversity cannot be maximized in accordance with different constellation sizes.
The second approach first constructs rotation matrices with two and three dimensions, which can be used as base matrices for constructing matrices with more dimensions using a Hadamard-like stacked expansion shown in the following Math. 11.
                    R        =                  (                                                                      +                                      R                    1                                                                                                -                                      R                    2                                                                                                                        +                                      R                    2                                                                                                +                                      R                    1                                                                                )                                    [                  Math          .                                          ⁢          11                ]            
The base 2-D and 3-D (three-dimensional) rotation matrices have a single independent parameter which is chosen so that the product distance of the constellation is maximized. A 4-D rotation matrix is constructed from two 2-D rotation matrices according to the above Math. 11. Because of the relative small dimension, it is possible to find an algebraic relationship between parameters of the two 2-D rotation matrices, so that the product distance is maximized. For larger dimensions, such an optimization becomes intractable, which is the primary disadvantage of the second approach.
(Mapping Constellation Components to Ensure Independent Fading)
Another aspect concerns the separation and mapping of the N dimensions of the rotated constellation so that they experience independent fading. This is a key aspect necessary for achieving the expected diversity performance.
The N constellation components, which are obtained by separating the N-dimensional rotated constellation on a per-dimension basis, can be transmitted over different time slots, frequencies, transmitter antennas, or combinations thereof. Further signal processing is possible before transmission. The critical aspect is that fading experienced by each of the N dimensions must be different from, or ideally uncorrelated with, fading experienced by any other one of the N dimensions.
The spreading of the N dimensions across different time slots, frequencies and antennas can be achieved for example through appropriate interleaving and mapping.
(Mapping Constellation Components to Transmitted Complex Cells)
Another aspect concerns the mapping of the N real dimensions of the rotated constellation to complex symbols for transmission. In order to ensure the desired diversity, the N dimensions must be mapped to different complex symbols. The complex symbols are then spread as described earlier, e.g. through interleaving and mapping, so that at the reception, fading experienced by each of the N dimensions is uncorrelated with fading of any other one of the N dimensions.
FIG. 2 is a block diagram of a transmission apparatus.
The transmission apparatus is composed of an FEC encoder 210, a bit interleaver 220, a rotated constellation mapper 230, a complex symbol mapper 240, a symbol interleaver/mapper 250, modulation chains 260-1 to 260-M, and transmitter antennas 270-1 to 270-M.
The FEC encoder 210 performs forward error correction (FEC) encoding on the input thereto. Note that the best FEC codes known so far, which are also the most used in new standards, are the turbo codes and the low-density parity check (LDPC) codes.
The bit interleaver 220 performs bit interleaving on the input from the FEC encoder 210. Here, the bit interleaving can be block interleaving or convolution interleaving.
The rotated constellation mapper 230 maps the input from the bit interleaver 220 to the rotated constellation.
Generally, the input to the rotated constellation mapper 230 is the output of the FEC encoder 210 via the bit interleaver 220 that performs optional bit interleaving. The bit interleaving is usually required when there are more than one bit per dimension (B>1). The FEC encoding performed by the FEC encoder 210 introduces redundant bits in a controlled fashion, so that propagation errors can be corrected in the reception apparatus. Although the overall spectral efficiency decreases, the transmission becomes overall more robust, i.e., the bit error rate (BER) decays much faster with the signal to noise ratio (SNR).
Note that regarding the original mapping of the information bits on the non-rotated hyper-cubic constellations, each dimension is modulated separately by B bits, using either binary or Gray mapping, so the number of discrete values is 2B and the number of constellation points is 2B*N.
The complex symbol mapper 240 maps each of N constellation components, which represent N-dimensional rotated constellation symbols input from the rotated constellation mapper 230, to a different one of complex symbols.
There are multiple possibilities for the mapping performed by the complex symbol mapper 240, i.e., the mapping of each of N constellation components, which represent N-dimensional rotated constellation symbols, to a different one of complex symbols. Some of such possibilities are illustrated in FIG. 3. The essential function of the complex symbol mapper 240 is to map each of N constellation components of one rotated constellation symbol to a different one of complex symbols.
By way of example, FIG. 3 shows the case of four dimensions. Referring to FIG. 3, the boxes showing the same number (e.g., “1”) represent a group of 4-D rotated constellation symbols. The number shown by each box indicates the group number of the corresponding group. Also, each box indicates a constellation component of one dimension.
Shown below “Constellation symbols” in FIG. 3 is a state where six groups of 4-D rotated constellation symbols are aligned. Shown below “Complex symbols” in FIG. 3 are twelve complex symbols, which are obtained by rearranging the six groups of 4-D rotated constellation symbols shown below “Constellation symbols” in FIG. 3. Note that FIG. 3 shows three forms of “Complex symbols” as examples. At the time of actual transmission, a pair of two constellation components that are vertically aligned below “Complex symbols” (the result of rearrangement) is modulated and transmitted as one complex symbol.
The symbol interleaver/mapper 250 performs symbol interleaving on the complex symbols input from the complex symbol mapper 240, and thereafter maps the complex symbols to different time slots, frequencies, transmitter antennas, or combinations thereof. Here, the symbol interleaving can be block interleaving or convolution interleaving.
The modulation chains 260-1 to 260-M are provided in one-to-one correspondence with the transmitter antennas 270-1 to 270-M. Each of the modulation chains 260-1 to 260-M inserts pilots for estimating the fading coefficients into the corresponding input from the symbol interleaver/mapper 250, and also performs various processing, such as conversion into the time domain, digital-to-analog (D/A) conversion, transmission filtering and orthogonal modulation, on the corresponding input. Then, each of the modulation chains 260-1 to 260-M transmits the transmission signal via a corresponding one of the transmitter antennas 270-1 to 270-M.
(Receiver Side)
On the receiver side, the exact inverse steps of the steps performed by the transmission apparatus must be performed. FIG. 4 shows a block diagram of a reception apparatus corresponding to the transmission apparatus whose block diagram is shown in FIG. 2.
The reception apparatus is composed of receiver antennas 410-1 to 410-M, demodulation chains 420-1 to 420-M, a symbol demapper/deinterleaver 430, a complex symbol demapper 440, a rotated constellation demapper 450, a bit deinterleaver 460, and an FEC decoder 470.
The demodulation chains 420-1 to 420-M are provided in one-to-one correspondence with the receiver antennas 410-1 to 410-M. Each of the demodulation chains 420-1 to 420-M performs processing such as A/D conversion, reception filtering, and orthogonal demodulation on the signal transmitted by the transmission apparatus of FIG. 2 and received by a corresponding one of the receiver antennas 410-1 to 410-M. Then, the demodulation chains 420-1 to 420-M estimate (i) the amplitude values (fading coefficients) of the channel characteristics by using the pilots and (ii) noise variance, and output the estimated amplitude values and noise variance together with the phase-corrected received signal.
The symbol demapper/deinterleaver 430 performs the inverse processing of the processing performed by the symbol interleaver/mapper 230 in the transmission apparatus on the inputs from the demodulation chains 420-1 to 420-M.
The complex symbol demapper 440 performs the inverse processing of the processing performed by the complex symbol mapper 240 in the transmission apparatus on the input from the symbol demapper/deinterleaver 430. Through this processing, N-dimensional rotated constellation symbols can be obtained.
The rotated constellation demapper 450 performs demapping processing on the N-dimensional rotated constellation symbols, and outputs a decision result of each bit included in the N-dimensional rotated constellation.
The bit deinterleaver 460 performs the inverse processing of the processing performed by the bit interleaver 220 in the transmission apparatus on the input from the rotated constellation demapper 450.
The FEC decoder 470 performs FEC decoding on the input from the bit deinterleaver 470.
Below, further explanations of the rotated constellation demapper 450 are given.
The rotated constellation demapper 450 can perform the processing of demapping N-dimensional rotated constellation symbols in the following two ways (i) and (ii).
(i) First de-rotate the constellation, then extract the bits for each dimension separately.
(ii) Decode the bits of all dimensions in one step.
Although the first solution (the above (i)) is the most simple, its performance is suboptimal and even worse for rotated constellations than for non-rotated constellations. Due to its simplicity, this solution may be used in some low-cost reception apparatuses.
Although the second solution (the above (ii)) is more complex, it offers much better performance in terms of BER at a given SNR. In the following, the second solution will be described in greater detail.
As with the transmission apparatus, a preferred embodiment of the reception apparatus includes the FEC decoder 470 after the rotated constellation demapper 450, with the optional bit deinterleaver 460 in between, as shown in FIG. 4. More exactly, the rotated constellation demapper 450, which performs the rotated constellation demapping, receives N-dimensional symbol vectors (y1, . . . , yN) and the estimated fading coefficient vectors (h1, . . . , hN), and extracts data of N*B bits (b1, . . . , bN*B) from each symbol, as shown in FIG. 5.
When FEC decoding is used, the processing of demapping the N-dimensional rotated constellation symbols can no longer be performed by way of a hard decision, because the performance of the error correction would be suboptimal. Instead, “soft bits” must be used, either in the form of probabilities or in the form of log-likelihood ratios (LLRs). The LLR representation is preferred because probability multiplications can be conveniently expressed as sums. By definition, the LLR of a bit bk is shown in the following Math. 12.
                              L          ⁡                      (                                          b                k                            ❘              y                        )                          ⁢                  =          △                ⁢                  ln          ⁢                                    P              ⁡                              (                                                      b                    k                                    =                                      1                    ❘                    y                                                  )                                                    P              ⁡                              (                                                      b                    k                                    =                                      0                    ❘                    y                                                  )                                                                        [                  Math          .                                          ⁢          12                ]            
Note that in Math. 12,                P(bk=0|y)and        P(bk=1|y)are the a-priori probabilities that bk=0 and bk=1 were transmitted when the symbol vector            is received. According to the known theory, the LLR of a bit bk of a constellation has the exact expression shown in the following Math. 13.
                              L          ⁡                      (                          b              k                        )                          =                  ln          ⁢                                                    ∑                                  s                  ∈                                      S                    k                    1                                                                                                                ⁢                                                          ⁢                              exp                ⁡                                  (                                      -                                                                                                                                                  y                            -                            Hs                                                                                                    2                                                                    2                        ⁢                                                                                                  ⁢                                                  σ                          2                                                                                                      )                                                                                    ∑                                  s                  ∈                                      S                    k                    0                                                                                                                ⁢                                                          ⁢                              exp                ⁡                                  (                                      -                                                                                                                                                  y                            -                            Hs                                                                                                    2                                                                    2                        ⁢                                                                                                  ⁢                                                  σ                          2                                                                                                      )                                                                                        [                  Math          .                                          ⁢          13                ]            
Note that in Math. 13, k is the bit index,    is the received symbol vector,    is the diagonal matrix having the associated (estimated) fading coefficients as elements on the main diagonal,    is a constellation point vector,            ∥∥2 is the squared norm, and        σ2 is the noise variance.        
For an N-dimensional constellation, the squared norm represents the squared Euclidean distance from the received symbol vector    to the faded constellation symbol vector    in the N-dimensional space. The squared norm can be expressed by the following Math. 14.
                                                                    y              -              Hs                                            2                =                              ∑                          n              =              1                        N                    ⁢                                          ⁢                                                                                    y                  n                                -                                                      h                    n                                    ⁢                                      s                    n                                                                                      2                                              [                  Math          .                                          ⁢          14                ]            
Each bit bk divides the constellation into two partitions of equal size, Sk0 and Sk1, corresponding to those points for which bk is 0 and 1, respectively. Examples are shown in FIGS. 6A and 6B for a classical 16-QAM constellation with Gray encoding. FIG. 6A shows the constellation encoding and FIG. 6B shows the two partitions for each bit bk.
The exact expression for the LLR (the above Math. 13) is difficult to calculate due to the exponentials, divisions and the logarithm. In practice, the approximation shown in the following Math. 15 is made, called max-log, which introduces negligible errors.ln(ea1+ea2)≈max(a1,a2)→ln(e−a1+e−a2)≈min(a1,a2)  [Math. 15]
By using the above Math. 15, the above Math. 13 leads to a much more simple expression for the LLR, which is shown in the following Math. 16.
                              L          ⁡                      (                          b              k                        )                          ≈                                            1                              2                ⁢                                                                  ⁢                                  σ                  2                                                      ⁢                                          min                                  s                  ∈                                      S                    k                    0                                                              ⁢                                                          ⁢                                                                                      y                    -                    Hs                                                                    2                                              -                                    1                              2                ⁢                                                                  ⁢                                  σ                  2                                                      ⁢                                          min                                  s                  ∈                                      S                    k                    1                                                              ⁢                                                          ⁢                                                                                      y                    -                    Hs                                                                    2                                                                        [                  Math          .                                          ⁢          16                ]            
For each received symbol vector    ,the distances to all 2B*N constellation points must be calculated, and the corresponding minimum for each partition is determined.
FIG. 7 shows a preferred hardware implementation of an LLR demapper (one example of the rotated constellation demapper 450 shown in FIG. 4) for a 16-QAM rotated constellation (N=2, B=2).
The LLR demapper is composed of a counter 710, a rotated constellation mapper 720, a squared Euclidean distance calculator 730, minimizers 740-1 to 740-4, and adders 750-1 to 750-4.
For each received symbol vector    ,the counter 710 repeatedly generates all 24=16 constellation points, and outputs four bits b1, b2, b3 and b4 indicating the constellation points to the rotated constellation mapper 720.
The rotated constellation mapper 720 selects the 2-D rotated constellation point from a look-up table by using the counter values provided by the counter 710 as an indexes, and outputs two constellation components s1 and s2 obtained through this selection to the squared Euclidean distance calculator 730.
The squared Euclidean distance calculator 730 calculates the squared Euclidean distances (see FIG. 8).
For each bit, the minimizers 740-1 to 410-4 maintain the corresponding minimum squared Euclidean distances for the two partitions (see FIG. 9). The two constellation partitions for each bit are simply indicated by the corresponding bit of the counter 710.
Each of the adders 750-1 to 750-4 subtracts the output of min1 (corresponding to bit 1) from the output of min0 (corresponding to bit 0), the min1 and min0 being provided in each of the minimizers 740-1 to 740-4. Thereafter, the adders 750-1 to 750-4 output the results of the subtraction as L(b1) to L(b4), respectively.
FIG. 8 is a circuit diagram of a squared Euclidean distance calculator that calculates an N-dimensional squared Euclidean distance. Note that the circuit structure of the squared Euclidean distance calculator 730 has been modified from the one shown in FIG. 8 so as to satisfy N=2.
The squared Euclidean distance calculator is composed of multipliers 810-1 to 810-N, adders 820-1 to 820-N, multipliers 830-1 to 830-N, an adder 840, and a multiplier 850.
The multipliers 810-1 to 810-N multiply h1 to hN by s1 to sN, respectively. The adders 820-1 to 820-N subtract h1s1 to hNsN from y1 to yN, respectively. The multipliers 830-1 to 830-N multiply (y1−h1s1) to (yN−hNsN) by (y1−h1s1) to (yN−hNsN), respectively.
The adder 840 adds together the outputs of the multipliers 830-1 to 830-N. The multiplier 850 multiplies the output of the adder 840 by    1/(2σ2).The output of the multiplier 850 is the N-dimensional squared Euclidean distance.
FIG. 9 is a circuit diagram of the minimizers 740-1 to 740-4 that each calculate the minimum squared Euclidean distances for each bit. The 1-bit subset (or partition) input indicates the current position.
Each of the minimizers 740-1 to 740-4 is composed of a comparator 910, a selector 920, an inverter 930, D flip-flops 940-0 and 940-1, and a selector 950.
The following describes the operations to be performed in the situation of FIG. 9 when the subset value (the value input from the counter 710) is “0”.
From among the output of the D flip-flop 940-0 and the output of the D flip-flop 940-1, the selector 950 selects and outputs the former.
The comparator 910 compares din (A), which indicates the squared Euclidean distance calculated by the squared Euclidian distance calculator 730, with the output (B) of the selector 950. In a case where B is smaller than A, the comparator 910 outputs “0”. In this case, from among din and the output of the selector 950, the selector 920 selects and outputs the latter based on “0” received from the comparator 910. On the other hand, in a case where A is smaller than B, the comparator 910 outputs “1”. In this case, from among din and the output of the selector 950, the selector 920 selects and outputs the former based on “1” received from the comparator 910. Note that in a case where A is equal to B, the same result will be obtained whether the selector 920 selects din or the output of the selector 950. Accordingly, in this case, the comparator 910 may output either one of “0” and “1”.
The inverter 930 inverts the subset value “0”. Thus, “1” is input to the enable terminal of the D flip-flop 940-0. As the D flip-flop 940-0 is enabled, it latches the output of the selector 920. Meanwhile, “0” is input to the enable terminal of the D flip-flop 940-1. As the D flip-flop 940-1 is disabled, it does not latch the output of the selector 920.
The following describes the operations to be performed in the situation of FIG. 9 when the subset value is “1”.
From among the output of the D flip-flop 940-0 and the output of the D flip-flop 940-1, the selector 950 selects and outputs the latter.
The comparator 910 compares din (A) with the output (B) from the selector 950. In a case where B is smaller than A, the comparator 910 outputs “0”. In this case, from among din and the output of the selector 950, the selector 920 selects and outputs the latter based on “0” received from the comparator 910. On the other hand, in a case where A is smaller than B, the comparator 910 outputs “1”. In this case, from among din and the output of the selector 950, the selector 920 selects and outputs the former based on “1” received from the comparator 910. Note that in a case where A is equal to B, the same result will be obtained whether the selector 920 selects din or the output of the selector 950. Accordingly, in this case, the comparator 910 may output either one of “0” and “1”.
“1” is input to the enable terminal of the D flip-flop 940-1. As the D flip-flop 940-1 is enabled, it latches the output of the selector 920. Meanwhile, the inverter 930 inverts the subset value “1”. Thus, “0” is input to the enable terminal of the D flip-flop 940-0. As the D flip-flop 940-0 is disabled, it does not latch the output of the selector 920.
A significant improvement in performance of the reception apparatus can be achieved by using iterative decoding. As shown in FIG. 10, the reception apparatus configured to utilize such iterative decoding is composed of a rotated constellation demapper 1010, a bit deinterleaver 1020, an FEC decoder 1030, an adder 1040, and a bit interleaver 1050. Here, the rotated constellation demapper 1010 and the FEC decoder 1030 are connected in a loop.
The rotated constellation demapper 1010 performs demapping processing on N-dimensional rotated constellation symbols, and outputs L (see FIG. 11). The bit deinterleaver 1020 performs the inverse processing of the processing performed by the bit interleaver 220 in the transmission apparatus on the input from the rotated constellation demapper 1010. The FEC decoder 1030 performs FEC decoding on the input from the bit deinterleaver 1020.
The adder 1040 subtracts the input from the FEC decoder 1030 from the output of the FEC decoder 1030. The bit interleaver 1050 performs the same processing as the processing performed by the bit interleaver 220 in the transmission apparatus on the output of the adder 1040, and then outputs LE. LE, also referred to as extrinsic information, is fed back to the rotated constellation demapper 1010 in order to aid the demapping processing performed by the rotated constellation demapper 1010, i.e., the processing of demapping the N-dimensional rotated constellation symbols. In this case it is essential that the FEC decoding produces soft bits, e.g. in the form of LLRs.
As known in the literature, the formula for calculating the LLR for bit bk is given by the following Math. 17.
                              L          ⁡                      (                          b              k                        )                          ≈                                            min                              x                ∈                                  X                  k                  0                                                      ⁢                          {                                                                    1                                          2                      ⁢                                              σ                        2                                                                              ⁢                                                                                                          y                        -                                                  Hs                          ⁡                                                      (                            x                            )                                                                                                                                      2                                                  +                                                      ∑                                                                  i                        =                        1                                                                                                                          x                            i                                                    =                          1                                                ,                                                  i                          ≠                          j                                                                                      K                                    ⁢                                                            L                      E                                        ⁡                                          (                                              b                        i                                            )                                                                                  }                                -                                    min                              x                ∈                                  X                  k                  1                                                      ⁢                          {                                                                    1                                          2                      ⁢                                              σ                        2                                                                              ⁢                                                                                                          y                        -                                                  Hs                          ⁡                                                      (                            x                            )                                                                                                                                      2                                                  +                                                      ∑                                                                  i                        =                        1                                                                                                                          x                            i                                                    =                          1                                                ,                                                  i                          ≠                          j                                                                                      K                                    ⁢                                                            L                      E                                        ⁡                                          (                                              b                        i                                            )                                                                                  }                                                          [                  Math          .                                          ⁢          17                ]            
In Math. 17,                represents the K=N*B bits associated with each constellation point, and Xk0 and Xk1 represent the two constellation partitions associated with bit k, each constellation point being represented by the N*B bits instead of the N bits of integer coordinates. Furthermore,        is expressed as        and represents the constellation mapping function.        
For example, X30 and X31 are shown in the following Math. 18.
                                                        X              3              0                                                          X              3              1                                                            0000                                0100                                                0001                                0101                                                0010                                0110                                                0011                                0111                                                1000                                1100                                                1001                                1101                                                1010                                1110                                                1011                                1111                                              [                  Math          .                                          ⁢          18                ]            
FIG. 11 shows an example of the structure of the rotated constellation demapper 1010 for iterative decoding. Note that the rotated constellation demapper 1010 for iterative decoding is similar to a rotated constellation demapper for non-iterative decoding. Below, the elements that are the same as those described above are assigned the same reference numerals thereas, and a detailed description thereof is omitted.
The rotated constellation demapper 1010 is composed of a counter 710, a rotated constellation mapper 720, a squared Euclidean distance calculator 730, minimizers 740-1 to 740-4, adders 750-1 to 750-4, logical AND operators 1110-1 to 1110-4, an adder 1120, adders 1130-1 to 1130-4, and adders 1140-1 to 1140-4.
The logical AND operators 1110-1 to 1110-4 perform logical AND operations on the outputs of the bit interleaver 1050, namely LE(b1) to LE(b4), and the outputs of the counter 710, namely b1 to b4. The adder 1120 adds together the outputs of the logical AND operators 1110-1 to 1110-4. Each of the adders 1130-1 to 1130-4 subtracts, from the output of the adder 1120, the output of a corresponding one of the logical AND operators 1110-1 to 1110-4. Each of the adders 1140-1 to 1140-4 subtracts, from the output of the squared Euclidean distance calculator 730, the output of a corresponding one of the adders 1130-1 to 1130-4. Then, each of the adders 1140-1 to 1140-4 outputs the value obtained through the subtraction to din of a corresponding one of the minimizers 740-1 to 740-4.